A mathematical model of multicomponent ion transport through a cation-exchange membrane is developed based on the Nernst-Planck equation. A correlation for the non-linear potential gradient is derived from current density relation with fluxes. The boundary conditions are determined with the Donnan equilibrium at the membranesolution interface, taking into account the convective flow. Effective diffusivities are used in the model based on the correlation of tortuosity and ionic diffusivities in free water. The model predicts the effect of an increase in current density on the ion concentrations inside the membrane. The model is fitted to the previously published experimental data. The effect of current density on the observed increase in voltage drop and the decrease in permselectivity has been analyzed using the available qualitative membrane swelling theories. The observed nonlinear behavior of the membrane voltage drop versus current density can be explained by an increase in membrane pore diameter and an increase in the number of active pores. We show how the membrane pore diameter increases and dead-end pores open up when the current density is increased.

This work describes a model for bilayer cationexchange membranes used in the chlor-alkali process. The ion transport inside the membrane is modeled with the Nernst-Planck equation. A logistic function is used at the boundary between the two layers of the bilayer membrane to describe the change in the properties of each membrane layer. The local convective velocity is calculated inside the membrane using the Schlögl equation and the equation of continuity. The model calculates the ion concentration profiles inside the membrane layers. Modeling results of mono-and bilayer membranes are compared. The changes in membrane voltage drop and sodium selectivity are predicted. The concentration profile of sodium ions in the bilayer membrane is significantly different from the monolayer membrane. Without the applied current, a linear change in the sodium concentration is observed in the monolayer membrane and in each layer of the bilayer membrane. With an increase in current density, the stronger electromotive force in the carboxylate layer causes a decrease in the sodium concentration in the sulfonate layer, down to the fixed ionic group concentration. This significant decrease of sodium ion concentration in the sulfonate layer results in low concentrations of counter ions and as a consequence a higher permselectivity of the bilayer membrane is obtained when compared to the single-layer membrane. As a drawback, the resistance in the bilayer membrane increases.